There are many mathematical proofs of this, which make perfect sense, but their connection is counter-intuitive as they perform seemingly different operations on curves. The derivative finds the slope at any point and the integral sums infinitesimal slabs of area under the curve. All derivatives can be solved analytically, while that is not true for integrals. Integration depends to an additive constant, while derivatives do not. Dissimilarities surely are numerous.
I have recently found this paper I wrote during my undergraduates in the quest to understand the geometrical origin of the connection between. Hope it is of any help to you, if you are having trouble in this like I had. I remember it made much more sense when I - very unrigorously - put it in this simple way:
P.S. for "Derivate" I mean "Derivative"
